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For a quantum error correcting code to be used in practice, it needs to beequipped with an efficient decoding algorithm, which identifies correctionsgiven the observed syndrome of errors.Hypergraph product codes are a promisingfamily of constant-rate quantum LDPC codes that have a linear-time decodingalgorithm called Small-Set-Flip ($\texttt{SSF}$) (Leverrier, Tillich, Z\'emorFOCS 2015). The algorithm proceeds by iteratively applying small correctionswhich reduce the syndrome weight. Together, these small corrections canprovably correct large errors for sufficiently large codes with sufficientlylarge (but constant) stabilizer weight. However, this guarantee does not holdfor small codes with low stabilizer weight. In this case, $\texttt{SSF}$ canterminate with stopping failures, meaning it encounters an error for which itis unable to identify a small correction. We find that the structure of errorsthat cause stopping failures have a simple form for sufficiently small qubitfailure rates. We propose a new decoding algorithm called theProjection-Along-a-Line ($\texttt{PAL}$) decoder to supplement $\texttt{SSF}$after stopping failures. Using $\texttt{SSF}+\texttt{PAL}$ as a combineddecoder, we find an order-of-magnitude improvement in the logical error rate.

Addressing Stopping Failures for Small Set Flip Decoding of Hypergraph Product Codes